Find ideal set and its monic generator

Question

It's my first time to learn ideal in linear algebra(H/K)

I wonder if my thought is right.

Question is

let $Q$ be the field of rational numbers. Determine which of the following subsets of $Q[x]$ are ideals. When the set is an ideal, find its monic generator.

for all f in the range of the linear operator T defined by

$T(\sum_{i=0}^n c_ix^i)=\sum_{i=0}^n \frac{c_i}{i+1}x^{i+1}$

I think it is ideal since monic generator is $x$ that make all $xQ[x]$ in Range $T$ and $x$ is in Range $T$ ($T(1)=x$ when $c_0=1$). Is it right?